January  2022, 21(1): 159-181. doi: 10.3934/cpaa.2021173

Marcinkiewicz integral operators along twisted surfaces

1. 

Department of Mathematics, Sultan Qaboos University, Sultanate of Oman

2. 

Department of Mthematics, Yarmouk University, Irbid, Jordan

* Corresponding author

Received  June 2021 Revised  September 2021 Published  January 2022 Early access  September 2021

Marcinkiewicz integral operators on product domains defined by translates determined by twisted surfaces are introduced. Maximal functions along twisted surfaces are also introduced. Conditions on the underlined surfaces implying that the corresponding Marcinkiewicz integral operators map $ L^{p}\rightarrow L^{p} $ for $ 1<p<\infty $ are obtained. A general method involving lacunary families of multi-indices is developed.

Citation: Ahmad Al-Salman. Marcinkiewicz integral operators along twisted surfaces. Communications on Pure and Applied Analysis, 2022, 21 (1) : 159-181. doi: 10.3934/cpaa.2021173
References:
[1]

H. Al-QassemA. Al-SalmanL. Cheng and Y. Pan, Marcinkiewicz integrals on product spaces, Stud. Math., 167 (2005), 227-234.  doi: 10.4064/sm167-3-4.

[2]

H. Al-Qassem and A. Al-Salman, Rough Marcinkiewicz integral operators, Int. J. Math. Math. Sci., 27 (2001), 495-503.  doi: 10.1155/S0161171201006548.

[3]

A. Al-SalmanH. Al-Qassem and Y. Pan, Singular integrals on product domains, Indiana Univ. Math. J., 55 (2006), 369-387.  doi: 10.1512/iumj.2006.55.2626.

[4]

A. Al-Salman, Singular integral operators on product domains along twisted surfaces, Front. Math. China, 16 (2021), 13-28.  doi: 10.1007/s11464-021-0911-z.

[5]

A. Al-Salman, Maximal functions along surfaces on product domains, Anal. Math., 34 (2008), 163-175.  doi: 10.1007/s10476-008-0301-8.

[6]

A. Al-Salman, Rough Marcinkiewicz integrals on product spaces, Int. Math. Forum, 2 (2007), 1119-1128.  doi: 10.12988/imf.2007.07097.

[7]

A. Al-Salman, Maximal operators with rough kernels on product domains, J. Math. Anal. Appl., 311 (2005), 338-351.  doi: 10.1016/j.jmaa.2005.02.048.

[8]

A. Al-Salman and H. Al-Qassem, Rough singular integrals on product spaces, Int. J. Math. Math. Sci., 67 (2004), 3671-3684.  doi: 10.1155/S0161171204312342.

[9]

A. Al-Salman and Y. Pan, Singular integrals with rough kernels in$Llog^{+}L(S^{n-1})$, J. London Math. Soc., 66 (2002), 153-174.  doi: 10.1112/S0024610702003241.

[10]

A. Al-SalmanH. Al-QassemL. Cheng and Y. Pan, $L^{p}$ boundes for the functions of Marcinkiewicz, Math. Res. Lett., 9 (2002), 697-700.  doi: 10.4310/MRL.2002.v9.n5.a11.

[11]

A. Al-Salman and H. Al-Qassem, Integral operators of Marcinkiewicz type, J. Integral Equ. Appl., 14 (2002), 343-354.  doi: 10.1216/jiea/1181074927.

[12]

A. BenedekA. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci., 48 (1962), 356-365.  doi: 10.1073/pnas.48.3.356.

[13]

J. ChenD. Fan and Y. Ying, Rough Marcinkiewicz integrals with $L(\log L)^{2}$kernels, Adv. Math. (China), 30 (2001), 179-181. 

[14]

Y. Choi, Marcinkiewicz integrals with rough homogeneous kernel of degree zero in product domains, J. Math. Appl., 261 (2001), 53-60.  doi: 10.1006/jmaa.2001.7465.

[15]

J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Ins. Fourier (Grenoble), 36 (1986), 185-206. 

[16]

J. Duoandikoetxea and J. L. Rubio de Francia, Maximal functions and singular integral operators via Fourier transform estimates, Invent. Math., 84 (1986), 541-561.  doi: 10.1007/BF01388746.

[17]

Y. Ding, L$^{2}$-boundedness of Marcinkiewicz integral with rough kernel, Hokkaido Math. J., 27 (1998), 105-115.  doi: 10.14492/hokmj/1351001253.

[18]

D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math., 119 (1997), 799-839. 

[19]

R. Feferman and E. M. Stein, Singular integrals on product spaces, Adv. Math., 45 (1982), 117-143.  doi: 10.1016/S0001-8708(82)80001-7.

[20]

F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals, J. Funct. Anal., 73 (1987), 179-194.  doi: 10.1016/0022-1236(87)90064-4.

[21]

E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proc. Int l. Cong. Math., (1986), 196–221.

[22]

E. M. Stein, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., 88 (1958), 430-466.  doi: 10.2307/1993226.

[23] E. M. Stein, Harmonic Analysis: Real-Variable Mathods, Orthogonality and Oscillatory integrals, Princeton University Press, Princeton, NJ, 1993. 
[24]

H. Wu, Boundedness of multiple Marcinkiewicz integral operators with rough kernels, J. Korean Math. Soc., 43 (2006), 635-658.  doi: 10.4134/JKMS.2006.43.3.635.

show all references

References:
[1]

H. Al-QassemA. Al-SalmanL. Cheng and Y. Pan, Marcinkiewicz integrals on product spaces, Stud. Math., 167 (2005), 227-234.  doi: 10.4064/sm167-3-4.

[2]

H. Al-Qassem and A. Al-Salman, Rough Marcinkiewicz integral operators, Int. J. Math. Math. Sci., 27 (2001), 495-503.  doi: 10.1155/S0161171201006548.

[3]

A. Al-SalmanH. Al-Qassem and Y. Pan, Singular integrals on product domains, Indiana Univ. Math. J., 55 (2006), 369-387.  doi: 10.1512/iumj.2006.55.2626.

[4]

A. Al-Salman, Singular integral operators on product domains along twisted surfaces, Front. Math. China, 16 (2021), 13-28.  doi: 10.1007/s11464-021-0911-z.

[5]

A. Al-Salman, Maximal functions along surfaces on product domains, Anal. Math., 34 (2008), 163-175.  doi: 10.1007/s10476-008-0301-8.

[6]

A. Al-Salman, Rough Marcinkiewicz integrals on product spaces, Int. Math. Forum, 2 (2007), 1119-1128.  doi: 10.12988/imf.2007.07097.

[7]

A. Al-Salman, Maximal operators with rough kernels on product domains, J. Math. Anal. Appl., 311 (2005), 338-351.  doi: 10.1016/j.jmaa.2005.02.048.

[8]

A. Al-Salman and H. Al-Qassem, Rough singular integrals on product spaces, Int. J. Math. Math. Sci., 67 (2004), 3671-3684.  doi: 10.1155/S0161171204312342.

[9]

A. Al-Salman and Y. Pan, Singular integrals with rough kernels in$Llog^{+}L(S^{n-1})$, J. London Math. Soc., 66 (2002), 153-174.  doi: 10.1112/S0024610702003241.

[10]

A. Al-SalmanH. Al-QassemL. Cheng and Y. Pan, $L^{p}$ boundes for the functions of Marcinkiewicz, Math. Res. Lett., 9 (2002), 697-700.  doi: 10.4310/MRL.2002.v9.n5.a11.

[11]

A. Al-Salman and H. Al-Qassem, Integral operators of Marcinkiewicz type, J. Integral Equ. Appl., 14 (2002), 343-354.  doi: 10.1216/jiea/1181074927.

[12]

A. BenedekA. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci., 48 (1962), 356-365.  doi: 10.1073/pnas.48.3.356.

[13]

J. ChenD. Fan and Y. Ying, Rough Marcinkiewicz integrals with $L(\log L)^{2}$kernels, Adv. Math. (China), 30 (2001), 179-181. 

[14]

Y. Choi, Marcinkiewicz integrals with rough homogeneous kernel of degree zero in product domains, J. Math. Appl., 261 (2001), 53-60.  doi: 10.1006/jmaa.2001.7465.

[15]

J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Ins. Fourier (Grenoble), 36 (1986), 185-206. 

[16]

J. Duoandikoetxea and J. L. Rubio de Francia, Maximal functions and singular integral operators via Fourier transform estimates, Invent. Math., 84 (1986), 541-561.  doi: 10.1007/BF01388746.

[17]

Y. Ding, L$^{2}$-boundedness of Marcinkiewicz integral with rough kernel, Hokkaido Math. J., 27 (1998), 105-115.  doi: 10.14492/hokmj/1351001253.

[18]

D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math., 119 (1997), 799-839. 

[19]

R. Feferman and E. M. Stein, Singular integrals on product spaces, Adv. Math., 45 (1982), 117-143.  doi: 10.1016/S0001-8708(82)80001-7.

[20]

F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals, J. Funct. Anal., 73 (1987), 179-194.  doi: 10.1016/0022-1236(87)90064-4.

[21]

E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proc. Int l. Cong. Math., (1986), 196–221.

[22]

E. M. Stein, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., 88 (1958), 430-466.  doi: 10.2307/1993226.

[23] E. M. Stein, Harmonic Analysis: Real-Variable Mathods, Orthogonality and Oscillatory integrals, Princeton University Press, Princeton, NJ, 1993. 
[24]

H. Wu, Boundedness of multiple Marcinkiewicz integral operators with rough kernels, J. Korean Math. Soc., 43 (2006), 635-658.  doi: 10.4134/JKMS.2006.43.3.635.

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